1Introduction DeterminantalRepresentationoftheTime-DependentStationaryCorrelationFunctionfortheTotallyAsymmetricSimpleExclusionModel

Determinantal Representation of the Time-Dependent Stationary Correlation Function for the Totally

Asymmetric Simple Exclusion Model

Nikolay M. BOGOLIUBOV

St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia

E-mail: [email protected]

Received October 30, 2008, in final form April 14, 2009; Published online April 23, 2009 doi:10.3842/SIGMA.2009.052

Abstract. The basic model of the non-equilibrium low dimensional physics the so-called totally asymmetric exclusion process is related to the ‘crystalline limit’ (q → ∞) of the SU_q(2) quantum algebra. Using the quantum inverse scattering method we obtain the exact expression for the time-dependent stationary correlation function of the totally asymmetric simple exclusion process on a one dimensional lattice with the periodic boundary conditions.

Key words: quantum inverse method; algebraic Bethe ansatz; asymmetric exclusion process 2000 Mathematics Subject Classification: 82C23; 81R50

1 Introduction

The study of quantum exactly solvable models within the framework of the quantum inverse scattering method (QISM) [1,2,3] has led to an algebraic structures which are q-deformations of the universal enveloping algebras [4, 5]. These deformed structures are usually called now

‘quantum groups’ or ‘quantum algebras’. Integrable models associated with the ‘crystalline limit’ (q → ∞) [6] of these algebras play an important role in enumerative combinatorics [7,8]

and non-equilibrium low dimensional physics [9,10].

The algebraic approach to the calculation of the correlation functions for the integrable models within the frames of the (QISM) was developed in the late eighties of the last century [11]

advantage of this approach is that the form of the final answers depends on theR-matrix of the model and not on the particular realization of the operators intertwained by this matrix. It is also important that the operator averages are calculated over the eigenstates of the generating function of the integrals of motion.

The correlation functions for the boson model associated with the ‘crystalline limit’ of the XXZ R-matrix were calculated in [12,13] where it was shown that they have the determinantal form.

The totally asymmetric simple exclusion process (TASEP) is an integrable model that is connected with the ‘crystalline limit’ of theXXZ R-matrix. It is one of the most studied models of the low dimensional non-equilibrium physics. The Bethe ansatz approach to the solution of the model was developed in the papers [14, 15, 16, 17, 18, 19]. The relation of the model to the QISM has been studied in [20, 21, 22], the model with the twisted boundary conditions within the frames of this method was considered in [23], and the asymmetric exclusion process in [24]. The dynamical properties of TASEP are intensively studied (see, e.g., [25,26,27] and

?This paper is a contribution to the Proceedings of the XVIIth International Colloquium on Integrable Sys- tems and Quantum Symmetries (June 19–22, 2008, Prague, Czech Republic). The full collection is available at http://www.emis.de/journals/SIGMA/ISQS2008.html

the references in these papers). The KPZ scaling limit of TASEP was considered, in particular, in [14,28,29].

In the present paper we shall consider the TASEP on a periodic lattice with the finite number of sites and with the finite number of particles. We shall represent the form-factors of the local operators in the determinantal form using the algebraic approach of the QISM. This result may be considered as the main result of the present paper.

The paper is structured as follows. In Section 2, the definition of the model is given. The algebraic approach to the solution of the model is presented in Section 3, in this section the stationary states are expressed through the state vectors of the model and the scalar product of the state vectors is expressed in the determinantal form. In Section 4, the calculation of the averages of the local projection operators over the state vectors of the model defined on a lattice withM sites is reduced to the calculation of the scalar product of the state vectors of the model defined on theM−1 sites. In Section5, the well known results on the Bethe ansatz solution are presented. The answer, based on the results of the section 6, for the time-dependent stationary correlation function is obtained. Finally, conclusive remarks are given in Section7.

2 Totally asymmetric simple exclusion model

The totally asymmetric exclusion process describes a system of N particles on a periodic ring withM sites labelledi=M, M−1, . . . ,2,1. The particles move randomly only in one direction.

We shall choose this direction from right to left. The exclusion rule forbids to have more than one particle per site.

This process can be conveniently represented using the spin description - in each lattice site the spin-up state corresponds to the empty site |0i, the spin-down state corresponds to the occupied one |1i. The configuration ofN particles (2N ≤M) located on the sites M ≥m₁ >

m₂>· · ·> m_N ≥1 is given by the basis vector

|m₁, m₂, . . . , m_Ni=σ_m⁻₁σ⁻_m2· · ·σ_m⁻_N|Ωi,

where the generating state |Ωi is the state with all spins up

|Ωi=⊗^M_i=1|0i_i.

The described process is defined by a non-Hermitian Hamiltonian [14]:

H =−

M

X

j=1

σ_j+1⁻ σ⁺_j + 1

4(σ_j+1^z σ_j^z−1)

. (1)

Here σ^±,z are the Pauli matrices, the matrix with subindex j acts nontrivially only in the j-th spin space of the total space of states of the chain C²⊗M

: sj =I⊗ · · · ⊗I ⊗s⊗I ⊗ · · · ⊗I, and the periodic boundary conditions are assumed: σ_n=σ_n+M.

In the matrix representation −H is the Markov matrix that controls the time evolution of the probability measure.

The first term of the Hamiltonian describes the hoppings of particles σ⁻_j+1σ⁺_j |0i_j+1|1i_j =|1i_j+1|0i_j.

The second term counts the number of the allowed jumps of the particles with hard-core repul- sion.

In our paper we shall consider the time-dependent stationary correlation function [14]

Z_N⁻¹hS_N|s₁e^−|t|Hsm|S_Ni, (2)

where the projection operator s_k= 1

2(1 +σ_k^z) (3)

has value 1 if there is no particle at sitek and 0 otherwise. In the steady state |S_Ni every spin configuration withN spins down has the equal weight:

|S_Ni= X

M≥m₁>m2>···>m_N≥1

|m₁, . . . , m_Ni.

This state is a ground state of Hamiltonian with the eigenvalue zero:

H|S_Ni= 0.

The left eigenvector ofH with the eigenvalue zero is

hS_N|= X

M≥m1>m2>···>m_N≥1

hm₁, . . . , mN|.

The total number of configurations in the ground state ZN =hS_N|S_Ni= M!

N!(M −N)!.

One can express the right steady state in the following form [30]:

|S_Ni=P_R^N|Ωi, (4)

where P_R^N is theN-th power of operator P_R=

M

X

k=1

sM· · ·s_k+1σ_k⁻,

s_j are projection operators (3) andσ_k⁻are Pauli matrices. Similarly for the left steady state one has

hS_N|=hΩ|P_L^N (5)

with the operator P_L equal to P_L=

M

X

k=1

σ_k⁺sk−1· · ·s1.

3 Solution of the model

To the solution of the model we shall apply the quantum inverse scattering method [1, 2, 3].

The L-operator of the considered model [21,23] is a 2×2 matrix with the entries acting on the space of states of an M-site spin-¹₂ chain:

L(n|u) =

us_n σ⁻_n σ_n⁺ uI−u⁻¹sn

=ssn+ (I−s) uI−u⁻¹sn

+σ⁻σ_n⁺+σ⁺σ_n⁻, (6) where the parameter u∈C.

TheL-operator (6) satisfies the intertwining relation R(u, v) (L(n|u)⊗L(n|v)) = (L(n|v)⊗L(n|u))R(u, v), in which R(u, v) is the 4×4 matrix

R(u, v) =









f(v, u) 0 0 0

0 g(v, u) 1 0

0 0 g(v, u) 0

0 0 0 f(v, u)









(7)

with

f(v, u) = u²

u²−v², g(v, u) = uv u²−v².

In the quantum group theory this R-matrix is known as the ‘crystal base’R-matrix [6].

The monodromy matrix is the product ofL-operators T(u) =L(M|u)L(M−1|u)· · ·L(1|u) =

A(u) B(u) C(u) D(u)

. (8)

The commutation relations of the matrix elements of the monodromy matrix are given by the sameR-matrix (7)

R(u, v) (T(u)⊗T(v)) = (T(v)⊗T(u))R(u, v). (9)

In the explicit form the most important of these relations for us are C(u)B(v) =g(u, v){A(u)D(v)−A(v)D(u)},

A(u)B(v) =f(u, v)B(v)A(u) +g(v, u)B(u)A(v), D(u)B(v) =f(v, u)B(v)D(u) +g(u, v)B(u)D(v),

[B(u), B(v)] = [C(u), C(v)] = 0. (10)

The transfer matrixτ(u) is the matrix trace of the monodromy matrix

τ(u) =u^−MtrT(u) =u^−M(A(u) +D(u)). (11)

The relation (9) means that [τ(u), τ(v)] = 0 for arbitrary values of parametersu,v.

The cyclic shift operatorτ in the total spin space C²⊗M

is expressed through the transfer matrix:

τ ≡τ(1) = Π₁₂Π₂₃· · ·ΠM−1M. (12)

Here

Πmn =smsn+ (I−sm)(I−sn) +σ_m⁻σ⁺_n +σ_m⁺σ_n⁻,

is the permutation operator: Π_mnσ_m =σ_nΠ_mn. The shift operator shifts the site indices τⁿ⁻¹σ1τ¹⁻ⁿ=σn

and possesses the property τ^M =I.

In terms of the transfer matrix (11) the Hamiltonian (1) is given by H =−1

2τ⁻¹(1) ∂

∂uτ(u)|_u=1. (13)

The right state vector of the model is the vector generated by the multiple action of operators B(u)˜ ≡u^−(M−1)B(u) on the generating state |Ωi=⊗^M_i=1|0i_i

|Ψ(u₁, u2, . . . , uN)i=

N

Y

i=1

B(u˜ i)|Ωi. (14)

The generating state is annihilated by the operator C(u) C(u)|Ωi= 0,

and it is an eigenvector of operators A(u) andD(u),

A(u)|Ωi=α(u)|Ωi, D(u)|Ωi=δ(u)|Ωi (15)

with the eigenvalues

α(u) =u^M, δ(u) = (u−u⁻¹)^M. The left state vector is equal to

hΨ(u₁, u₂, . . . , u_N)|=hΩ|

N

Y

i=1

C(u˜ _i), (16)

where ˜C(u_i) =u^−(M−1)C(u), andhΩ|B(u) = 0.

From definitions (6) and (8) one finds by a direct calculation thatu^M−1B(u) andu^M−1C(u) are polynomials in u² of degreeM−1:

u^M−1B(u) =u^2(M−1)P_R+· · ·+ (−1)^M−1σ_M⁻sM−1· · ·s1, u^M−1C(u) =u^2(M−1)P_L+· · ·+ (−1)^M−1s_M· · ·s₂σ₁⁺.

This decomposition allows us to express the steady states (4) and (5) through the state vec- tors (14) and (16), namely

|S_Ni= lim

{u}→∞

N

Y

i=1

B(u˜ i)|Ωi, hS_N|= lim

{u}→∞hΩ|

N

Y

i=1

C(u˜ i). (17)

The scalar product of the state vectors (14) and (16) may be evaluated directly by means of the commutation relations (10) with the help of Laplace formula for the determinant of the sum of two matrices. It also may be obtained as a special limit of the general formula for the scalar products (IX.6.26) of [3]. It should be mentioned that for theR-matrix (7) the auxiliary quantum fields (IX.6.24) commute and the answer for the scalar product has the determinantal form. It means that the scalar products of the state vectors of the integrable models associated with theR-matrix (7) are expressed as the determinants of the matrices with the entries which depend on f and g elements ofR-matrix and on the eigenvaluesα and δ (15) that characterize the particular model.

For the arbitrary variables u, v ∈ C the scalar product of the state vectors of the model is given by the following expression

hΨ(v₁, v₂, . . . , v_N)|Ψ(u₁, u₂, . . . , u_N)i

=







N

Y

j=1

1 (vjuj)^M−1

Y

N≥j>k≥1

vjvk

v²_k−v_j² Y

N≥l>n≥1

ulun

u²_l −u²_n







detQ. (18)

The entries of theN ×N matrixQare Qjk =

(

v^M_j uk−u⁻¹_k M uk

vj

N−1

−u^M_k vj−v_j⁻¹M uk

vj

−N+1)

1

u_k vj −

u_k vj

−1.

Using this representation and equalities (17) that represent the steady states through the state vectors of the model we can calculate the projection of the state vectors (14) and (16) on the steady states. For the left steady state we have

hS_N|Ψ(u₁, u2, . . . , uN)i= lim

{v}→∞

N

Y

i=1

hΨ(v₁, v2, . . . , vN)|Ψ(u₁, u2, . . . , uN)i. (19) This limit may be calculated with a help of the formula

vlimj→v

det{Φ(v_j, u_k)}

∆(v) = det

1 j!

∂^j

∂v^jΦ(v, u_k)

,

where Φ(v, u) is an arbitrary differentiable for at leastN times function of two variables, and

∆(v) is the Vandermonde determinant. Taking the limit in (19) we obtain [31]:

hS_N|Ψ(u₁, u2, . . . , uN)i=

N

Y

k=1

u²_k Y

N≥l>n≥1

1

u²_l −u²_ndetV^(M), where V^(M) is aN ×N matrix with the entries equal to

V_jk^(M)=

j−1

X

n=0

(−1)ⁿ M

n

u^2(j−1−n)_k , 1≤j≤N−1,

V_{N k}^(M)=−

M

X

n=N−1

(−1)ⁿ M

n

u^{−2(n−N+1)}_k . (20)

The projection hΨ(u₁, u2, . . . , uN)|S_Ni is given by the similar expression hΨ(u₁, u2, . . . , uN)|S_Ni=

N

Y

k=1

u²_k Y

N≥n>l≥1

1

u²_l −u²_ndet ˜V^(M), V˜_jk^(M)=

N−j

X

n=0

(−1)ⁿ M

n

u^2(N−j−n)_k , 2≤j≤N, V˜_1k^(M)=−

M

X

n=N

(−1)ⁿ M

n

u^{−2(n−N+1)}_k .

4 Form-factors

The calculation of the correlation function (2) we start with the calculation of the matrix ele- ments hS_N|s₁|Ψ(u₁, u₂, . . . , u_N)i and hΨ(v₁, v₂, . . . , v_N)|s₁|S_Ni which by means of (17) are ex- pressed through the form-factor of the projection operator, namely

hS_N|s₁|Ψ(u₁, u₂, . . . , u_N)i= lim

{v}→∞hΨ(v₁, v₂, . . . , v_N)|s₁|Ψ(u₁, u₂, . . . , u_N)i, hΨ(v₁, v2, . . . , vN)|s₁|S_Ni= lim

{u}→∞hΨ(v₁, v2, . . . , vN)|s₁|Ψ(u₁, u2, . . . , uN)i. (21)

By the definition (8) the monodromy matrix may be expressed as T(u) =

AM−1(u) BM−1(u) CM−1(u) DM−1(u)

us1 σ⁻₁ σ₁⁺ uI−u⁻¹s1

.

From this formula it follows that the entries of the monodromy matrix may be expressed as A(u) =uAM−1(u)s₁+BM−1(u)σ⁺₁,

B(u) =AM−1(u)σ₁⁻+uBM−1(u)−u⁻¹BM−1(u)s1, C(u) =uCM−1(u)s₁+DM−1(u)σ⁺₁,

D(u) =CM−1(u)σ⁻₁ +uDM−1(u)−u⁻¹DM−1(u)s1.

Making use of the equalitiess²_j =s_j,s_jσ_j⁻=σ⁺_j s_j = 0 we obtain, in particular, that

s₁B(u) = (u−u⁻¹)BM−1(u)s₁, C(u)s₁ =us₁CM−1(u). (22) Substituting the explicit form of the state vectors (14) and (16) into the form-factor (21) and taking into account the commutation relations (22) we find that for the arbitrary values of parameters u, v the form-factor is proportional to the scalar product (18) of the state vectors on a lattice with M−1 sites:

hΨ(v₁, v2, . . . , vN)|s₁|Ψ(u₁, u2, . . . , uN)i

=hΩ|

N

Y

j=1

C(v˜ _j)s₁

N

Y

i=1

B(u˜ _i)|Ωi=hΩ|

N

Y

j=1

C(v˜ _j)s²₁

N

Y

i=1

B(u˜ _i)|Ωi

=

N

Y

k=1

1−u⁻²_k hΩ|

N

Y

j=1

C˜M−1(vj)

N

Y

i=1

B˜M−1(ui)|Ωi.

Taking the limit {v} → ∞we obtain hS_L|s₁|Ψ(u₁, u2, . . . , uN)i= lim

{v}→∞

N

Y

i=1

hΨ(v₁, v2, . . . , vN)|s₁|Ψ(u₁, u2, . . . , uN)i

=

N

Y

k=1

1−u⁻²_k

{v}→∞lim hΩ|

N

Y

j=1

C˜M−1(v_j)

N

Y

i=1

B˜M−1(u_i)|Ωi

=

N

Y

k=1

u²_k−1 Y

N≥l>n≥1

1

u²_l −u²_ndetV^(M−1), (23)

where the entries of N ×N matrix V^(M−1) are (20) with M replaced on M −1. Respectively we have

hΨ(u₁, u₂, . . . , u_N)|s₁|S_Ni=

N

Y

k=1

u²_k Y

N≥n>l≥1

1

u²_l −u²_ndet ˜V^(M−1). (24)

5 Bethe ansatz solution

By following through the arguments of the algebraic Bethe ansatz method [1,2,3] one proves that the state vectors (14) and (16) are the right and the left eigenstates of the transfer matrix (11) with the same eigenvalues

τ(v)|Ψ(u₁, u2, . . . , uN)i= ΘN(v,{u})|Ψ(u₁, u2, . . . , uN)i,

hΨ(u₁, u2, . . . , uN)|τ(v) =hΨ(u₁, u2, . . . , uN)|Θ_N(v,{u}) (25) if allowed parameters u₁, u₂, . . . , u_N in (25) satisfy Bethe equations [14,15,16,17,18,19]:

1−u⁻²_n −M

u^−2N_n = (−1)^N−1

N

Y

j=1

u⁻²_j ≡(−1)^N−1U⁻². (26)

The eigenvalues Θ_N(v,{u}) are equal to Θ_N(v,{u}) =

N

Y

j=1

u²_j

u²_j −v² + 1−v⁻²M N

Y

j=1

v² v²−u²_j.

From the definition of the cyclic shift operator (12) it follows that its eigenvalues are equal to Θ_N(1,{u}) =

N

Y

j=1

1

1−u⁻²_j . (27)

From (13) one finds the eigenenergies of the Hamiltonian (1) E_N =−1

2Θ⁻¹_N (1,{u}) ∂

∂vΘ_N(v,{u})|_v=1 =−

N

X

j=1

1 u²_j −1.

The obvious solution u₁ = u₂ = · · · = u_N = ∞ provides the stationary solution with the eigenvalue E_N = 0. The steady state |S_Ni is the eigenvector of the transfer matrix (11) τ(u) with the eigenvalue equal to one:

τ(u)|S_Ni= Θ_N(v,{∞})|S_Ni=|S_Ni, and hence it is stochastic.

The scalar product of eigenvectors (14), (16) is found from formula (18) withvj =ujsatisfying Bethe equations (26). Understanding the diagonal elements of the matrix Q in the sense of L’Hˆopitall rule one obtains the following expression for the norm of any eigenvector:

N²(u1, u2, . . . , uN) =hΨ(u₁, u2, . . . , uN)|Ψ(u₁, u2, . . . , uN)i=U^2NY

l6=n

1

u²_l −u²_ndet ˜Q (28) with the entries of the matrix ˜Q equal to

Q˜_jk = N −1 + (M−N + 1)u⁻²_j

1−u⁻²_j δ_jk−(1−δ_jk).

For the special solution u1 =u2=· · ·=uN =∞ the norm of eigenvectors is equal toZN. There is Z_N solutions of the Bethe equations [16], and the state vectors belonging to the different sets of the solutions of the Bethe equations are orthogonal. The eigenvectors (25) provide the resolution of the identity operator

I =X

{u}

|Ψ(u₁, u2, . . . , uN)ihΨ(u₁, u2, . . . , uN)|

N²(u₁, u₂, . . . , u_N) ,

where the summation is over all different solutions of Bethe equations (26).

6 Stationary correlation function

For the transitionally invariant system the form-factor of the projection operators_mis expressed through the form-factor of the operator s1:

hΨ(u₁, u₂, . . . , u_N)|s_m|S_Ni=hΨ(u₁, u₂, . . . , u_N)|τ^m−1s₁τ^1−m|S_Ni

=

N

Y

j=1

1−u⁻²_j 1−m

hΨ(u₁, u2, . . . , uN)|s₁|S_Ni, where the property (27) was used.

The substitution of the resolution of the identity intohS_N|s₁e^−|t|Hsm|S_Nigives hS_N|s₁e^−|t|Hsm|S_Ni=X

{u}

hS_N|s₁e^−|t|H|Ψ(u₁, u2, . . . , uN)ihΨ(u₁, u2, . . . , uN)|s_m|S_Ni N²(u₁, u₂, . . . , u_N)

=X

{u}

e^−|t|ENhS_N|s₁|Ψ(u₁, u₂, . . . , u_N)ihΨ(u₁, u₂, . . . , u_N)|s₁|S_Ni N²(u₁, u₂, . . . , u_N)

N

Q

j=1

1−u⁻²_j m−1 . From the determinantal representations of the matrix elements (23), (24) and of the norm (28) we finally obtain the answer for the stationary correlation function (2):

1

Z_NhS_N|s₁e^−|t|Hs_m|S_Ni=

M −N M

2

+X

{u}

e^−|t|EN U^2N

N

Y

j=1

(u²_j −1)u²_j

1−u⁻²_j m−1

det ˜V^(M−1)detV^(M−1)

det ˜Q ,

where the summation is taken over all different solutions of Bethe equations (26) except the special one. The first term on the r.h.s. is the contribution of the stationary state.

7 Conclusive remarks

The models connected with the ‘crystal base’R-matrix (7) naturally appear in the theory of the boxed plane partitions – three-dimensional Young diagrams placed into a box of a finite size, and in the theory of random walkers [32, 8]. Some aspects of these connections were discussed in the papers [33,34,35,36,37,38].

In its turn plane partitions and random walkers are employed in analyzes of the models of statistical physics describing faceted crystals [39], direct percolation [40], one-dimensional growth processes [41]. It emphasizes the importance of the mentioned integrable models in the theory of the non-equilibrium processes.

Acknowledgements

The work was partially supported by the RFBR grant 07-01-00358.

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